Thirteenth USA/Europe Air Traffic Management Research and Development Seminar (ATM2019)

A Spectral Approach TowardsAnalyzing Air Traffic Network Disruptions

Max Z. Li, Karthik Gopalakrishnan, Hamsa BalakrishnanDepartment of Aeronautics and Astronautics

Massachusetts Institute of TechnologyCambridge, MA, USA

{maxli, karthikg, hamsa}@mit.edu

Kristyn PantojaDepartment of Statistics

Texas A&M University, College Station, TX, USAkristynp@stat.tamu.edu

Abstract—The networked nature of the air transportationsystem leads to systemwide delays and cancellations as a resultof disruptions at an airport. A comprehensive analysis of systemperformance requires understanding the inherent interdepen-dencies between various airports, in order to characterize off-nominal disruptions and to aid in recovery. In this work, weapply Graph Signal Processing (GSP) techniques to the analysisof flight delay networks, yielding two novel contributions: (1) Weuse the notion of the total variation (TV) of a graph signal inorder to identify and quantify unexpected distributions of delaysacross airports; and (2) we present a spectral eigendecompositionanalysis of airport disruption and delay networks. We investigateand characterize different patterns of delay distribution basedon the relationship between TV and total delay, using 10 yearsworth of operational data from major US airports. We showthat attributes of the resultant eigenvector modes and energycontributions are useful metrics to characterize specific disrup-tions caused by events such as nor’easters, Atlantic hurricanes,and equipment outages at airports.

Keywords: Graph Signal Processing; graph Fourier analysis;spectral methods; weather disruptions; airport operations; flightdelays

I. INTRODUCTION

The scale of air transport operations conducted within theUS National Airspace (NAS), along with the interconnected-ness inherent within the NAS contribute to the challenges incharacterizing the resilience and predictability of the system[1]. Obtaining a better understanding of these attributes are ofparticular importance in the context of disruptive events thataffect the NAS, leading to systemic penalties such as conges-tions, delays, and flight cancellations. The importance of thisline of research has been highlighted both domestically as wellas internationally; the Federal Aviation Administration (FAA)and EUROCONTROL have identified the need for increasedresiliency (how tolerant, or robust, airport-airspace operationsare to operational perturbations) and predictability (reductionin variability and uncertainty) within the air transportationsystem.

Weather disruptions were responsible for 53% of all de-lays experienced in the NAS in 2017 [2]. One particular

This work was supported in part by the US National Science Foundation.

meteorological phenomenon we examine in this work is thenor’easter, a type of winter storm that predominantly affect theMid-Atlantic and New England region of the US. Nor’easterssignificantly disrupt air traffic operations; the nor’easter thatoccurred 22-24 January 2016 resulted in the cancellation ofover 11,000 flights, and was estimated to lead to financiallosses of $100 to $120 million USD to the US aviation industry[3]. Airport and airline equipment outages also lead to delaysand cancellations. We examine two particular cases involvinga power outage at Hartsfield-Jackson Atlanta InternationalAirport (ATL) and a fire incident at the Chicago Air RouteTraffic Control Center (ZAU); both resulted in hundreds tothousands of cancelled flights throughout the NAS [4, 5].

Furthermore, since each aircraft typically operates morethan one flight per day, it is well-known that both air traffic andcrew-induced delays tend to propagate throughout the system(Section II-A). Moreover, combined with the increasing trendof airline hub strengthening at a handful of major airports, theshare of traffic – and consequently, delays – is being handledby a smaller subset of increasingly critical and interconnectedhub airport nodes. Due to the underlying networked structureof the NAS, any analysis of airport disruptions that do notconsider the connectivities and interactions between airportsmay result in myopic findings. To this end, we propose usingGraph Signal Processing (GSP) techniques (Section II-B)to analyze the impact and dynamic signatures of differentcategories of disruptions at major US airports. We presentinsights into the delay impact of different types of disruptiveevents (e.g. Atlantic hurricanes) and the nature of recoveryby analyzing the spectrum of the airport delay signals on theunderlying graph network.

A. Motivation: Delays in a network

High delays within the NAS are known to have significanteconomic impacts. When analyzing system performance in-volving multiple airports and routes, delay can be measuredin several ways. Since the airport is typically the bottleneck ofthe system and the location where delay statistics are recorded,airport-centric delay metrics are commonly used. For instance,

some classic metrics to quantify the performance of an airportuse the average delay of departing flights, or the total delay ofall the departing flights. However, due to the strong underlyingnetwork connectivity, some airport delays are highly correlatedto other. This correlation leads to patterns of delay propagationthat are familiar to airport operators, airlines, and air trafficcontrollers.

Within the NAS, system-wide disruptions can take on twoforms: (1) The obvious problematic scenario when delays arehigh across all airports; (2) a subtler scenario when delaysare high, but distributed only across a limited set of airports.The former can be identified via classical metrics, but thelatter is more difficult to discern. There are also importantoperational implications of the second scenario. Consider anexample involving Boston Logan International Airport (BOS)and LaGuardia Airport (LGA) in New York. Historically,delays at BOS and LGA are strongly correlated due to factorssuch as their geographical proximity and connectedness viatraffic flows. This means that when delays are high at one ofthe two airports, it is likely that the other airport will be ina state of high delays as well. Operationally, this relationshipis significant and may warrant air traffic controllers to coor-dinate Traffic Management Initiatives such as Ground DelayPrograms and En Route Separation Programs to alleviate thedelays at both BOS and LGA, providing spatial and temporalbuffers for recovery. However, if there are high delays at onlyone of the two airports, it would constitute an unexpected delaypattern, potentially requiring a different approach towardsrecovery at the network-level.

We would like to define the terms off-nominal, expected,and unexpected in the context of our work. Continuing theexample above with BOS and LGA, envision the followingtwo scenarios: (1) BOS and LGA both have 90 minutes ofdelay; (2) BOS has 10 minutes of delay whereas LGA has170 minutes. In both scenarios, a singular or a series ofoff-nominal events (e.g. pop-up thunderstorms, terminal areacongestion) resulted in delays at BOS and LGA. However,the first scenario is an expected distribution of delays whereasthe second scenario is an unexpected one given historicallyhigh correlations of delays at BOS and LGA. In short, themagnitude of delays determines whether the system is nominalor off-nominal, whereas the distribution of delays determineswhether the system is expected or unexpected. Operationally,the two scenarios can have very different impacts, requiredifferent TMIs, and necessitate different airline recovery tac-tics. This example shows precisely why classical metrics failin these circumstances. To address this knowledge gap, ourwork focuses on how graph-theoretic and spectral methods canprovide insights to these unexpected and off-nominal events.

A priori knowledge that delays at two airports are notstrongly correlated can also be important operationally: Thelack of delay correlations provide a natural decomposition ofnetwork control decisions that may advocate for operationallyindependent recovery plans for these two airports. An examplein this case would be Phoenix Sky Harbor International Air-port (PHX) and Detroit Metropolitan Wayne County Airport

(DTW), an airport pair that we found to have very lowdelay correlations. It is important to identify and quantifysuch historical delay dynamics in order to improve opera-tional response and recovery planning. The main challengelies in identifying the expected performance with a networkperspective, where delays at multiple airports simultaneouslyaffect each other. Another challenge is defining what expectedpatterns are, keeping in mind that even high delays, if presentin a consistent pattern, could be considered expected. Finally,an open question in airline network operations is identifyingfundamental differences in delay dynamics caused by one typeof event (e.g., a nor’easter) as opposed to another (e.g., anAtlantic hurricane), even when the resultant total delays arethe same. Our goal is a data-driven analysis at the networklevel in order to (1) identify off-nominal days during whichunexpected delay dynamics occurred, (2) discuss the opera-tional implications, and (3) demonstrate the value of such ananalysis using case studies.

II. LITERATURE REVIEW

A. Delays and disruptions

Due to the structure of air transportation networks, localairport-level disruptions may spread to other airports andpersist in the system. The complex dynamics of delays areprimarily due to aircraft rotations, wherein a single aircraftflies multiple trips in a given day between airport pairs.Consequently, the delays at two airports are not indepen-dent, and depend on the traffic between the airports, airlinescheduling and crew rotation policies, geographic proximity,airline networks, and air traffic control procedures. Severalapproaches, including queueing theory [6], network models[7], simulations [8], and machine learning methods [9], havebeen employed to understand delay dynamics.

Localized and NAS-wide disruptions cause delays and flightcancellations requiring airlines to adjust their flight schedules,re-assign crews, and accommodate stranded passengers. Typi-cally, off-nominal events in the NAS are identified by lookingat delay statistics at airports or on specific routes. Recently,clustering has been proposed as a tool for network-levelidentification of similar delay patterns [10, 11]. However, thesenetwork-level identification methodologies do not distinguishnor decompose the observed delay patterns into componentsthat occur often (and therefore, would be expected to occur),and ones that are more unusual (the unexpected components).This paper addresses this open question.

B. Graph Signal Processing (GSP) and graph Fourier decom-position

A foundational tool in classical signal processing is theFourier transform, an integral transform that decomposes atime-based signal into its components in the frequency domain.The Fourier transform is used in a wide selection of appli-cations ranging from signal de-noising and reconstruction tocircuit analysis. Fourier transforms have been previously usedin aviation applications, including the modeling and predictionof aircraft trajectories on final approach [12], the projection of

large trajectory data sets into feature spaces that can be easilyclustered and compressed [13], and the analysis of airportcapacity and delay profiles [14, 15]. This body of prior workalso considered the relationship between individual airportsin terms of delay propagation [15] . However, the underlyinggraph structure of the NAS – a critical component contributingto the behavior of delays and inefficiencies – cannot be takeninto account by the standard Fourier transform.

Given the natural occurrence of graphs and graph signals inaviation, we use GSP techniques to analyze airport disruptionsdue to weather and other outages. Our work builds off ofprevious applications of spectral and graph wavelet analysison traffic congestion in networks [16, 17]. Continuous andspectral graph wavelet transforms have also been used tostudy air traffic flows [18, 19], albeit at the resolution ofAir Route Traffic Control Centers (ARTCCs) and with limitedinterpretability of spectral results. We extend these approachesby focusing on origin-destination airports. We focus on delaysrather than traffic, and provide theoretical results that mapspectral analysis to operational interpretations.

III. METHODOLOGY

We first present the graph-theoretic notations and basicdefinitions, followed by Proposition 1 in which we show therelationship between total variation and the graph spectraldecomposition. We then formally define the notions of signalsmoothness across a graph, setting the stage for Proposi-tion 2 which relates correlation networks to smoothness andunexpected events. We provide explicit probabilistic boundson unexpected events given our setup. Propositions 1 and 2are the core of our theoretical contributions, and enable theinterpretability of our results when we apply GSP techniquesto airport delay networks. Due to the limitations of space, weomit the proofs of these propositions in this paper; we referinterested readers to [20] for proofs and additional theoreticalwork on identifying outliers in graph signals.

A graph G is defined by a set of nodes N containing |N |=N nodes, and edges E where an edge ei, j ∈ E connects apair of nodes (ni,n j) ∈N ×N . The weights on the edgesare represented by an adjacency matrix A ∈ RN×N , where theelement ai j is the weight on the edge from node ni to noden j. We consider only undirected graphs, which implies thatA = Aᵀ. The degree of a node ni is given by

deg(ni) =N

∑j=1

ai j. (1)

The degree matrix D is a diagonal matrix with elementsdii = deg(i). Given a graph G along with its adjacency matrixA and degree matrix D, we can define the combinatorialgraph Laplacian, L = D−A. The graph Laplacian L is a realsymmetric matrix with a full set of orthonormal eigenvectors.We denote the Laplacian eigenvectors by vi ∈ RN×1 fori ∈ {1, ...,N}. By orthogonality, we also have that vᵀi v j = δi j,where δi j = 1 if i = j, and 0 otherwise.

The corresponding eigenvalues λi are given by solutions toLvi = λivi. The eigenvalues are sorted such that λ1≤ λ2≤ . . .≤

λN . By construction, the rows of the Laplacian sum to zero,indicating that λ1 = 0 is always an eigenvalue. The multiplicityof λ1 is the number of connected components of the graph.Thus, for a fully connected graph, we would have 0 = λ1 <λ2 ≤ λ3 ≤ . . .≤ λN .

The set of eigenvectors {v1, . . . ,vN} form the graph Fourierbasis for any graph signal x∈RN×1 on the nodes of the graph;we will refer to x as a nodal graph signal, or simply a graphsignal with the understanding that it is supported on the nodesof the graph. The graph Fourier transform of a graph signalx is given by (α1, . . . ,αN), where each αi can be computed as

αi = xᵀvi. (2)

Intuitively, the graph Fourier transform (also known asspectral decomposition) quantifies the “contribution” of eacheigenvector in the graph Fourier basis to x.

The total energy of the signal x is defined as ‖x‖2, whichis also equal to ∑i α2

i . The energy spectrum of a signal,(α2

1 , ...,α2N), quantifies the distribution of the energy across

different graph eigenvectors. α2i is also referred to as the

energy contribution of the ith mode to the graph signal x.

Definition 1 (Total variation (TV)). The total variation (TV)of a graph signal x on the graph G = (N ,E ,A) is given by:

TV = xᵀLx =12 ∑

i6= jai j(xi− x j)

2. (3)

A graph signal with lower TV is said to be smoother relativeto another graph signal with a higher TV.

More precise definitions of graph signal smoothness aremade in Definitions 2 and 3.

A. Relationship between TV and spectral decomposition

Proposition 1. The following two statements are equivalent:(i) Total variation (TV) of a signal x is xᵀLx

(ii) Let Lvi = λivi, for i = 1, ...,N, 〈vi,v j〉 = δi j, and x =

∑i αivi, for some scalars αi. Then the TV of x is ∑i α2i λi.

B. Correlations, smoothness, and unexpected events

Given two nodes ni,n j ∈ N connected by an edgeei, j = {ni,n j} ∈ E , observe k samples of the paired data{(

x(1)ni ,x(1)n j

), ...,

(x(k)ni ,x

(k)n j

)}where x(`)n{i, j} is the `th observa-

tion of the pair of random nodal signals on nodes ni and n j,for `= 1, ...,k. Let a pair of observations

(x(`)ni ,x

(`)n j

)be drawn

from the bivariate Gaussian distribution given by(X (`)

ni

X (`)n j

)∼N

(µ

µ

), σ2

1 ρX(`)

ni ,X(`)n j

ρX(`)

ni ,X(`)n j

1

. (4)

The Pearson correlation coefficient for random variables X (`)ni

and X (`)n j is ρ

X(`)ni ,X(`)

n j. We will now formalize the elements

ai j of the adjacency matrix A by defining them explicitly interms of a weight map. Let w : E → [0,1] : ei, j 7→ w(ei, j) =

ai j = r(k)xni ,xn jbe the weight map that maps an edge to the

projected sample Pearson correlation coefficient for a signalon node ni and n j computed from k prior data observations{(

x(1)ni ,x(1)n j

), ...,

(x(k)ni ,x

(k)n j

)}as

r(k)xni ,xn j= max

0,

k

∑`=1

(x(`)ni − µ̂i

)(x(`)n j − µ̂ j

)√

k

∑`=1

(x(`)ni − µ̂i

)2√

k

∑`=1

(x(`)n j − µ̂ j

)2

. (5)

We use the notations µ̂i, µ̂ j and σ̂2i , σ̂

2j to represent the in-

sample mean and variance. Furthermore, r(k)xni ,xn jis an estimate

(formally, a statistic) for the population Pearson correlationcoefficient ρ

X(`)ni ,X(`)

n j. We also formalize the notion of smooth-

ness by relating it to the TV for graph signals:

Definition 2 (Absolute smoothness). A signal(xni ,xn j

)on two

nodes ni and n j is said to be smooth with respect to the edgeei, j with weight w(ei, j) = r(k)xni ,xn j

if the contribution to the TV

of the graph signal approaches 0, i.e. r(k)xni ,xn j

(xni − xn j

)2→ 0.

Since the pair of signals(xni ,xn j

)is being drawn from the

joint distribution given in Equation (4), we modify the defi-nition of smoothness to a weaker, probabilistic version. Thiscan be thought of as the data-driven definition of smoothness:

Definition 3 (Probabilistic smoothness). A signal(xni ,Xn j |(Xni = xni)

)on two nodes ni and n j at the `th

observation is said to be smooth in probability with respect tothe edge ei, j with weight w(ei, j) = r(k)xni ,xn j

if, given observationxni , the contribution to the TV of the graph signal convergesto 0 in probability, i.e. r(k)xni ,xn j

(xni −Xn j |(Xni = xni)

)2 p⇀ 0.

The notationp⇀ indicates a convergence in probability.

The notation Xn j |(Xni = xni) indicates that Xn j is beingdrawn from the conditional distribution of Xn j given an ob-servation xni . Probabilistic smoothness can be interpreted asfollows: It is highly probable that the contribution to the TV ofthe graph signal approaches 0. Our main proposition relatingsmoothness and nodal signal correlation is as follows:

Proposition 2 (Smoothness and nodal signal correlation). Thefollowing remarks relating graph signal smoothness via TVand nodal signal correlation hold true:

(i) Given two weakly correlated nodal signals (r(k)xni ,xn j→ 0),

the graph signal supported by nodes ni,n j and edge ei, jis absolutely smooth.

(ii.a) Given two strongly correlated nodal signals (r(k)xni ,xn j→ 1)

and an observation xni , the graph signals supported bynodes ni,n j and edge ei, j is smooth in probability if andonly if

∥∥xni −Xn j |(Xni = xni)∥∥

1p⇀ 0.

(ii.b) Given two strongly correlated nodal signals (r(k)xni ,xn j→

1), the graph signals supported by nodes ni and n j, andedge ei, j is not smooth with probability 1− p, where pis defined as

p = Pr(xni −S ≤ Xn j ≤ xni +S |Xni = xni

)(6)

where S = tp,k−2σ̂ j

√1+ 1

k +(xni−µ̂i)

2

(k−1)σ̂2i

is half ofthe width of the (100p)% prediction interval forXn j |(Xni = xni), with tp,k−2 as the (100p)th-quantile ofa t-distribution with (k−2) degrees of freedom.

IV. DATA PROCESSING AND SETUP

We consider the top airports (FAA Core 30) in terms ofpassenger and airline activity in the US for our analysis. Weextract the total hourly departure and arrival delays at eachof these 30 airports for the period 2008-2017. The signalvector for each day x is obtained by summing over the hourlydeparture and arrival delays from 0000Z to 2300Z.

ATLBOSBWICLTDCADTWEWR

FLLIADJFKLGA

MCOMIAPHLPHXTPA

DFWIAH

MDWMSPORDDENSLCLASLAXPDXSANSEASFOHNL

AT

LB

OS

BW

IC

LTD

CA

DT

WE

WR

FL

LIA

DJF

KL

GA

MC

OM

IAP

HL

PH

XT

PA

DF

WIA

HM

DW

MS

PO

RD

DE

NS

LC

LA

SL

AX

PD

XS

AN

SE

AS

FO

HN

L

0.00

0.25

0.50

0.75

1.00

PearsonCorr.

Total Delay Adjacency Matrix

Fig. 1: Heat map of the adjacency matrix displaying thedelay correlation between the top 30 airports, computed fromEquation (5).

Fig. 2: Network with the top 30 airports as nodes and delaycorrelation as edge weights. Higher edge weights are alsoshown with a wider line.

We construct a graph using these 30 airports as nodes.The edge weight between any two nodes of the graph isthe correlation coefficient between the delays at the twoairports. We use FAA ASPM data for September 2018 toobtain the edge weights via Equation (5). The set of nodes

and edge weights forms the airport-delay correlation graph,on whose nodes we observe an airport performance signal xfor each day. Figure 1 displays the value of the correlationcoefficient between different airports. The same informationis presented with geographical context in Figure 2, which hasthe 30 airports as nodes, and the edge weight representing thecorrelation.

The highest correlations are observed between the airportsin the Northeast region. This means that the delays in these air-ports tend to move in the same direction. High traffic betweenairports in the Northeast region is one contributing factorto high correlations. Another instance of a high correlationpair of airport is ORD and MDW. While there is no trafficbetween these two airports (they are less than 20 miles apart),they almost always have the same weather impact. However,geographical proximity alone does not lead to similar delaytrends, as the network connectivity of the two airports maybe completely different. For instance, San Diego (SAN) andLos Angeles (LAX) are about 120 miles apart, and yet have amuch lower correlation than LAX and Salt Lake City (SLC),which are 750 miles apart.

V. IMPLICATIONS OF PROPOSITIONS 1 AND 2 FOR NASDELAY NETWORKS

In this section, we aim to provide intuition for our theoreti-cal results by interpreting TV and the spectral decompositionin the context of airport delays. The two main ideas we wishto convey are: (1) TV is a useful metric to help identifywhen a certain spatial distribution of delays across the airtransportation network is unexpected; (2) Unexpected delayscan be analyzed and interpreted by examining its spectralsignature. Proposition 2 quantifies the likelihood of observinga certain spatial distribution of delays using the TV as a metric,whereas Proposition 1 relates the TV of the airport delays tothe graph spectrum and identifies the specific patterns that areunexpected.

The key setup that enables our analysis is the edge weightbeing the correlation of delays between two airports. If delaysat two highly-correlated airports are very close to each other,then r(k)xni ,xn j

(xni − xn j

)2 is small, since(xni − xn j

)is small (case

(ii.a) of Proposition 2). If an airport pair is uncorrelated, thenr(k)xni ,xn j

is small, so the product r(k)xni ,xn j

(xni − xn j

)2 is smallregardless of the values of xni and xn j (case (i) of Proposition2). Note that the term r(k)xni ,xn j

(xni − xn j

)2 is the mutual pairwisecontribution of airport nodes ni and n j to the TV.

Recall that since TV is the sum of the termsr(k)xni ,xn j

(xni − xn j

)2 over all airport node pairs ni and n j, ifall airport node pairs have delays that follow expected trends(i.e., the airport delays on a particular day follow historicalcorrelations), then the TV is small. On the other hand, whenthe airport delays at historically highly-correlated airports donot follow typical correlations on a particular day, the signalwould have a high TV. The spatial distribution of delays acrossairports is said to be smooth if the TV is small (Definitions 2and 3). Thus, spatial delay distributions at airports that followhistorical trends tend to be smooth.

vvviii ARTCC(s) Trend 1 Trend 2v1 Constant Constant Constant

v2ZAB, ZLA, ZJX,

ZMA, ZOA, ZSE, ZTL MCO, FLL, CLT SFO, SAN, ABQLAS, PDX

v3 ZJX, ZLA, ZOA, ZTL CLT MCO, SFO, LAX

v4ZAB, ZJX, ZLA

ZMA, ZTL FLL, ABQ CLT, MCO, LAX

v5

ZAU, ZBW, ZDC, ZDV,ZFW, ZJX, ZLA, ZLC,ZMA, ZNY, ZOA, ZOB

ZSE, ZTL

PDX, LAS, SANDEN, SLC, SFO

All other airportsin the ARTCCs (∗)

v6ZDV, ZLA, ZMA

ZOA, ZSE DEN LAS, PDX, MIASFO

v7 ZDV, ZLA, ZSE PDX LAS, SAN, DEN

v8ZBW, ZDC, ZLAZMA, ZNY, ZSE PDX, FLL, LAX Same as (∗)

v9ZJX, ZLA, ZLC

ZMA, ZOA, ZSE, ZTL MIA, SAN, SLC LAS, SFO, ATLMCO

v10ZDV, ZJX, ZLA, ZMA,

ZOA, ZSE, ZTLSFO, SEA, MIA

DEN Same as (∗)

v11

ZBW, ZDC, ZDVZFW, ZHU, ZJXZLA, ZMA, ZMP

ZNY, ZOA, ZOB, ZTL

FLL, MCO, SFOSAN, DEN Same as (∗)

v12 ZJX, ZMA, ZTL ATL, FLL MCO, CLT

v13

ZAU, ZDC, ZFW, ZHU,ZJX, ZLA, ZLC,

ZMA, ZMP, ZNY, ZOA,ZOB, ZSE, ZTL

ATL, MCO, SFOSAN Same as (∗)

v14ZAU, ZDV, ZHUZLA, ZMA, ZOA LAS, DEN MIA, IAH, ORD

SFO

v15

ZAU, ZBW, ZDC, ZFW,ZHU, ZJX, ZMP, ZNY,

ZOB, ZSE, ZTL

CLT, BWI, IADSEA Same as (∗)

v16ZDV, ZLA, ZMA

ZSE, ZTL ATL, DEN MIA, LAS, SEA

v17

ZAU, ZBW, ZDC, ZFW,ZHU, ZJX, ZMP, ZNY,

ZOB, ZSE, ZTL

MSP, CLT, SEAATL, DFW Same as (∗)

v18

ZAU, ZBW, ZDC, ZFW,ZHU, ZJX, ZMP,

ZNY, ZOB, ZSE, ZTL

ORD, MDW, DTWSEA, CLT, ATL Same as (∗)

v19ZBW, ZDC, ZFW, ZJX,ZLC, ZMP, ZNY, ZTL TPA, CLT Same as (∗)

v20

ZAU, ZBW, ZDCZFW, ZHU, ZJXZMP, ZNY, ZSE

IAH, DFW, ORD Same as (∗)

v21ZDC, ZDV, ZFW, ZLA,ZJX, ZMA, ZOB, ZTL

ATL, LAS, TPAMIA

DFW, DEN, DTWIAD

v22ZAU, ZBW, ZDCZFW, ZHU, ZNY DFW, MDW Same as (∗)

v23ZAU, ZBW, ZDC

ZNY, ZOB MDW Same as (∗)

v24ZAU, ZBW, ZDC

ZNY, ZOB BWI, MDW Same as (∗)

v25 ZBW, ZDC, ZNY IAD Same as (∗)

v26 ZBW, ZDC, ZNY LGA, IAD JFK, EWR, PHLDCA

v27 ZBW, ZDC, ZNY DCA, PHL IAD, BWI, JFKv28 ZBW, ZDC, ZNY BOS, EWR JFK, LGA, DCAv29 ZBW, ZDC, ZNY EWR BOSv30 ZBW, ZDC, ZNY PHL Same as (∗)

TABLE IDESCRIPTION OF EIGENVECTORS, DELAY TRENDS (TRENDS 1 AND 2

MOVE IN OPPOSITE DIRECTIONS), AND THE IMPACTED AIRPORTS ANDARTCCS. FIGURE 5 PROVIDES A VISUALIZATION OF v24 AND v29 .

As an illustrative example, take a pair of airports withperfectly correlated delays (i.e. their correlation coefficient is1). The contribution to the TV from this pair of airports isalways 0, since any new delays at both airports would beequal to each other with probability 1. However, in reality,such correlations may be close to, but not equal to, 1. Thus,given delays at one airport, the delays at the other cannot be

deterministically known and instead follow some probabilitydistribution. Case (ii.b) of Proposition 2 characterizes thisdistribution and specifies a prediction interval for the secondairport’s delay (with probability p). For airport pairs with ahigher correlation, the interval width would be smaller. In otherwords, the difference between the delays at both airports wouldbe bounded by an interval of width 2×S (Equation 6 in case(ii.b) of Proposition 2) with probability p. The contributionof this airport pair to the TV would be small with probabilityp. On the other hand, with probability 1− p, the airport pairwould have an unexpected distribution of delays, resulting ina larger contribution to the TV.

In practice, there are many airport pairs with a correlationcoefficient between 0 and 1. In such cases, they will contributeto the TV whenever their delays are not the same. Thus, itis important to remember that even an expected distributionof delays would result in a non-zero TV. Furthermore, withincreasing delays at airports, the TV is expected to growquadratically (Equation 3). Therefore, it is important to onlydraw comparisons of TVs between days with similar quantitiesof delay. With these considerations, it is possible to concludewhether or not a distribution of delays is unexpected (i.e. notsmooth) with respect to historical patterns.

While Proposition 2 allows for an airport-pair-specific de-composition of TV and interpretation of unexpected events,Proposition 1 provides a more global interpretation via spectraldecomposition (Equation 2). The graph Fourier decompositionof airport delays into eigenvector modes provides a set of“templates” or spatial delay patterns (see Table I) in theform of eigenvectors that can be used to reconstruct anyparticular day’s delay patterns across our airport graph. Theseeigenvector modes represent a set of airport delay patternseach with varying abilities to contribute to the TV (Proposition1). Furthermore, the contribution of each eigenvector mode tothe TV is proportional to its eigenvalue. Due to case (ii.b) ofProposition 2, delay patterns with higher TV are less likelyto occur. Proposition 1 tells us that having more energy athigher eigenvector modes leads to a higher TV. Conversely,when higher TVs are observed, there is more energy in thehigher eigenvector modes. This duality is highlighted by ourcase studies in Table II.

In Figure 3, we plot the total delay and corresponding TVfor each day in the US NAS from 2008 to 2017. This plotenables us to visualize and interpret Proposition 2. The red lineis the best quadratic fit for the data, and the dashed red lineshows the 99% prediction interval for the quadratic fit. Recallthat TV is expected to grow quadratically with total delaydue to Equation 3. However, days with a significantly higheror lower TV for a given level of total delay are unexpected;the delays on those days were distributed across airports in amanner not seen previously. In the 10-year period we analyzed,291 days (7.9%) fell outside of the 99% prediction interval.

It is worth noting that these days with unexpected spatialdistribution of delays are not the same as days with hightotal delays. Furthermore, unexpected days (in terms of TV)may or may not also have high delay; one attribute does not

imply the other, and vice versa. Specifically, what characterizesunexpected days is the fact that a very unusual and unexpectedspatial distribution of delays were observed across the airportswithin the network.

28 Dec 20102 Feb 2015

27 Jan 2011

3 Feb 2014

8 April 2016

17 Dec 2016

Fig. 3: TV versus Delay plot for 2008-2017.

Fig. 4: TV versus Delay plot for specific types of disruptions(nor’easters, hurricanes, airport equipment outages, and NAS-wide disruptions, in 2008-2017.

VI. ANALYSIS OF SPECIFIC OFF-NOMINAL EVENTS

We chose four categories of off-nominal events that havethe potential to cause severe disruptions: Atlantic hurricanes,nor’easters, airline-specific outages, and NAS-wide days whenthe entire system experiences significant delays, classified by[10]. Each event is recorded in Table II on a per-day basis. For

each day of a specific off-nominal event, we compute the totaldelay, the TV, and the graph Fourier decomposition. We alsocompute the energy contribution of each of the 30 eigenvectormodes.

For each day, we selected the most energy-contributingeigenvectors and constructed a set {v1, ...vi} of eigenvectors,ordered by their decreasing energy contribution. Typically, theconstant eigenvector v1 contains the majority of the energy.Using a threshold of 80% of the total spectrum energy,we construct the minimal set of eigenvectors such that thetotal energy of that set of eigenvectors meets or exceeds thethreshold. We selected the 80% energy threshold in order toretain the most important eigenvector modes while maintaininginterpretability. Each row in Table II corresponds to a particu-lar day of an off-nominal event; we give the eigenvector indexi ∈ {1, ...,30} as well as its individual energy contribution asa percentage.

A. Hurricanes

While hurricanes are known to cause major disruptionsfor airlines and airports, our spectral analysis shows thathurricane-induced disruptions tend to result in spatial delaydistributions with lower TV across our system. In Figure4, we see that days during three major Atlantic hurricanes(Sandy in October 2012, Harvey in end of August/beginningof September 2017, and Irma in September 2017) clustertowards the lower end both in terms of total delay and TV.More relevant in the context of the interplays between TV,smoothness, and unexpected events, the TVs exhibited byhurricanes do not exceed the 99% prediction interval. Thisindicates that, at least for the three hurricane events weexamine, the airport delay distributions across the system arenot considered to be unexpected.

This noticeable lack of graph signal variation for hurricane-related disruptions could be explained by a combination ofoperational and airline policy-driven responses specific to hur-ricanes. Typically, the most probable impact areas for Atlantichurricanes are limited to the Southern and occasionally Mid-Atlantic portions of the US. These projected impact areas arecontinuously updated as the hurricane approaches, allowingairlines to preemptively cancel and re-position flights andcrews. For example, once Hurricane Sandy was projected tomake landfall on the East Coast, more than 7,000 flights werecanceled ahead of Hurricane Sandy’s East Coast landfall onOctober 29, 2012 [21]. This action may have reduced theTV induced by Hurricane Sandy’s impacts due to the non-activation of East Coast modes (eigenvectors v25 through v30,see Table I). The highly-correlated airports on the East Coastdid not experience unexpected differences in their total delays.

We can also observe some interesting temporal trends inboth total delay and TV for disruptions caused by hurricanes.For all three hurricanes, the recovery phase (i.e. the last one ortwo days of the off-nominal event) is typically characterized bya drop in both total delay and TV. In the context of Proposition2, this trend suggests that the recovery phase towards theend of a hurricane’s impact is smooth and as expected.

This distinguishes hurricanes from nor’easters, where the TVtowards the end of a nor’easter may increase dramatically(e.g. the January 2016 nor’easter), even though the total delaycontinues to dissipate. In other words, our spectral analysissuggests that airline recovery responses towards hurricanesversus nor’easters may differ in nontrivial ways.

B. NAS-wide high delay days

Recall that these days were selected from clusters identifiedin [10] wherein the overall delays experienced by airportswithin the NAS were large. Even within this small selection ofNAS-wide high delay days, there are some interesting observa-tions. For example, the total delays on January 3 and January 5,2014 are comparable (4.61×104 versus 4.58×104 minutes),but the TV exhibited within our delay graph on January 3is almost twice as high (146.17× 106 versus 75.39× 106

minutes squared). The TV exhibited in the delay graph ofthe former day exceeds the 99% prediction interval of theTV versus total delay curve, whereas the latter day does not.In the context of Proposition 2, the delay graph exhibited onJanuary 3 represents an unusual event, even if the total delaywas comparable to the total delay two days later.

A large contributing factor resulting in the unusual delaygraph on January 3 is the activation of eigenvector v26, whereindelays at LGA and IAD are not trending with delays at JFK,EWR, PHL, and DCA, even though their delay signals arehistorically highly correlated. This is a highly energetic mode,providing a large contribution to the TV. Other highly energeticmodes such as v23 and v24 – the latter of which was alsoactivated on January 3 – capture delay trends at BWI andMDW moving out of sync with other major airports in ZAU,ZBW, ZDC, ZNY, and ZOB ARTCCs. In contrast, we only seethe activation of a medium-energy eigenvector v17 on January5 – the behavior of this mode summarized in Table I suggeststhat airlines with hubs at MSP, SEA and ATL, or at CLT andDFW may be operating in a way that causes the delays atthese five airports to not trend with other major airports in thelisted ARTCCs for v17.

C. Nor’easters

As mentioned earlier, nor’easters are known for severelyimpacting aviation operations, particularly around the EastCoast and Mid-Atlantic regions. The severity of these off-nominal events often forces airlines to preemptively cancelflights in order to lessen the propagative effects of moretactical delays and cancellations, as well as to better positionaircraft and crews for a quicker recovery.

From Table II, we note that nor’easter-type off-nominalevents tend to activate many eigenvectors that encapsulatevery high variations, i.e. vi’s where i ≥ 20. In particular,eigenvectors v24 and v29 (visualized in Figure 5) appear 7 and 5times, respectively. We can further augment our analysis froma geographic perspective by observing which airport nodesactivate in the two prevalent eigenvectors v24 and v29. Recallthat all East Coast and Mid-Atlantic airports tended to bestrongly positively correlated in terms of total delay (Figures

Event Date 1st 2nd 3rd 4th 5th Total Delay Total Variation(×104min) (×106min2)

Hurricane 10/28/12 1 (83%) – – – – 1.51 7.0110/29/12 1 (77%) 18 (4%) – – – 0.89 3.0110/30/12 1 (77%) 9 (4%) – – – 1.11 5.2310/31/12 1 (86%) – – – – 1.24 4.0411/1/12 1 (87%) – – – – 1.45 4.57

Hurricane 8/24/17 1 (87%) – – – – 1.63 4.018/25/17 1 (88%) – – – – 1.63 2.988/26/17 1 (58%) 20 (35%) – – – 1.39 19.848/27/17 1 (74%) 20 (13%) – – – 1.59 12.038/28/17 1 (78%) 18 (4%) – – – 1.65 10.218/29/17 1 (91%) – – – – 1.50 3.768/30/17 1 (91%) – – – – 1.26 1.66

Hurricane 9/9/17 1 (85%) – – – – 1.06 2.419/10/17 1 (77%) 19 (4%) – – – 0.96 2.999/11/17 1 (47%) 13 (17%) 12 (16%) 11 (4%) – 1.43 22.429/12/17 1 (89%) – – – – 1.39 2.89

NAS-wide 1/2/14 1 (68%) 23 (9%) 17 (7%) – – 4.33 113.441/3/14 1 (71%) 26 (6%) 24 (5%) – – 4.61 146.17

NAS-wide 1/5/14 1 (78%) 17 (4%) – – – 4.58 75.391/6/14 1 (79%) 5 (4%) – – – 3.87 41.92

NAS-wide 6/17/15 1 (67%) 20 (21%) – – – 2.19 33.39Nor’easter 2/25/10 1 (69%) 24 (9%) 5 (5%) – – 2.29 35.08

2/26/10 1 (60%) 24 (10%) 26 (5%) 5 (4%) 29 (4%) 2.89 90.302/27/10 1 (87%) – – – – 1.69 5.67

Nor’easter 1/30/11 1 (87%) – – – – 1.14 2.951/31/11 1 (74%) 6 (8%) – – – 1.64 9.692/1/11 1 (71%) 22 (10%) – – – 2.82 44.382/2/11 1 (72%) 24 (5%) 28 (4%) – – 2.25 32.252/3/11 1 (82%) – – – – 1.85 8.84

Nor’easter 2/7/13 1 (80%) – – – – 1.59 7.662/8/13 1 (86%) – – – – 1.73 8.512/9/13 1 (65%) 26 (11%) 29 (7%) – – 1.50 23.172/10/13 1 (75%) 15 (7%) – – – 1.64 11.332/11/13 1 (72%) 5 (5%) 27 (5%) – – 2.13 26.23

Nor’easter 2/11/14 1 (88%) – – – – 1.63 3.642/12/14 1 (76%) 13 (6%) – – – 2.31 16.872/13/14 1 (70%) 24 (5%) 5 (5%) – – 3.48 74.232/14/14 1 (86%) – – – – 2.82 15.87

Nor’easter 1/26/15 1 (74%) 5 (6%) – – – 1.94 16.681/27/15 1 (83%) – – – – 1.18 5.491/28/15 1 (84%) – – – – 1.28 4.181/29/15 1 (89%) – – – – 1.49 3.881/30/15 1 (79%) 24 (7%) – – – 2.05 14.15

Nor’easter 1/21/16 1 (90%) – – – – 1.72 4.551/22/16 1 (80%) – – – – 2.37 15.451/23/16 1 (58%) 29 (13%) 28 (8%) 12 (3%) – 1.61 36.301/24/16 1 (56%) 24 (11%) 29 (6%) 30 (6%) 25 (5%) 1.85 50.931/25/16 1 (68%) 29 (12%) 24 (5%) – – 1.79 28.60

Outage 11/15/12 1 (93%) – – – – 1.55 2.18Outage 9/26/14 1 (73%) 18 (10%) 2.33 25.71Outage 9/17/15 1 (80%) – – – – 1.73 8.02Outage 7/20/16 1 (84%) – – – – 2.05 11.26

7/21/16 1 (77%) 18 (9%) – – – 2.74 26.50Outage 8/8/16 1 (70%) 13 (7%) 12 (5%) – – 2.72 31.35

8/9/16 1 (87%) – – – – 2.23 6.69Outage 1/22/17 1 (79%) 13 (8%) – – – 3.09 16.66Outage 12/17/17 1 (65%) 13 (12%) 12 (10%) – – 1.64 14.86

TABLE IIDIFFERENT OFF-NOMINAL EVENTS; COLUMNS “1ST ” THROUGH “5TH ” CONTAIN THE HIGHEST-CONTRIBUTING EIGENVECTORS AND THEIR ENERGY

CONTRIBUTION, IN DESCENDING ORDER.

1 and 2), reflecting both the operational interconnectednessas well as the geographic proximity between these airports.Hence, the delay distribution is expected to be smooth (case(ii.a) of Proposition 2). However, we note that during theoff-nominal conditions imposed by various nor’easters, the

delays at BWI are not behaving in an expected mannerwhen compared to other East Coast airports (eigenvector v24).Similarly, the delays at BOS and EWR are not behaving in theexpected pairwise manner, as represented by eigenvector v29.Thus, our spectral analysis suggests that nor’easters, more so

Fig. 5: Eigenvectors v24 (top) and v29 (bottom).

than other off-nominal events, drive the NAS into historicallyunexpected modes.

Even within nor’easter-type events, there are interestingdifferences in the temporal distribution of the high-energycontributing eigenvectors. For example, the February 2010nor’easter peaks on February 26, driving the NAS into avery unexpected delay mode requiring five eigenvectors tocharacterize, three of which contain high levels of graph signalvariance (i∈{24,26,29}). However, an abrupt recovery occursbetween February 26 and 27, with the NAS settling back toa state where delay signals are smooth across the entire NAS(i.e. only v1 is required to fulfill the 80% energy threshold)in just one day. This does not appear to be the case for theFebruary 2013 nor the January 2016 nor’easter, where therecovery process was much less smooth in the context ofgraph signals. We hypothesize that this observation points toa non-trivial difference in recovery strategies beween differentnor’easters – some strategies result in a more “even” network-wide recovery, whereas others result in recoveries that areless smooth and isolated. Future research that attempts to linkdifferent operational features in airline irregular operations(IROPs) recovery strategies with excited eigenvector modeswould be of interest.

D. Airport- and airline-specific outages

Our spectral analysis shows that the airport- and airline-specific outages examined in Table II did not produce un-expected delay signals across our graph. Many of these off-nominal days (November 15, 2012; September 17, 2015; July20, 2016; August 9, 2016) did not require additional eigen-vectors past v1 to pass the 80% energy threshold, indicatinglow levels of TV in the airport delays. Even though the totaldelay in the system was at a significant level for some of thesedays (e.g. 2.05 ×104 and 2.23 ×104 minutes of delay for July20, 2016 and August 9, 2016, respectively), the TV was low

and well within the 99% prediction interval that delineates anexpected versus unexpected spatial delay distribution.

Furthermore, even when additional eigenvectors are requiredto achieve the 80% energy threshold, the eigenvectors tend tobe less energetic, providing less contribution to the TV. Forexample, the two outages on August 8, 2016 and December17, 2017 suffered by Delta Air Lines due to power outages atATL identically triggered eigenvector modes v12 and v13. Justfrom the index of the eigenvectors we can see that these arelow-contributing modes to the TV. Both modes record delaysignals at ATL trending opposite to delay signals at otherairports within the NAS, but the delays at these other airportsare weakly correlated to ATL (i.e. r(k)xni ,xn j

→ 0), thus explainingthe low contributions to the TV. If we examine Figures 1 or 2,we see that delay signals at ATL are more strongly correlatedwith East Coast airports such as BOS, DCA, EWR, LGA,and PHL, but these were not the airports whose delay signalswere trending opposite to ATL’s delay signals in eigenvectormodes v12 and v13 (case (i) of Proposition 2). Recall thatthese eigenvector modes have airport-specific geographicalinterpretations; v12 and v13, as described in Table I, implicatedelays at ATL trending opposite to delays at other southeastairports such as MCO and CLT.

Operationally, it may be the case that airport- and airline-specific outages cause severe disruptions, but these disruptionsare mostly localized to one particular airport node. In addition,for the cases of airport- and airline-specific outages that weexamine, the airport node that was affected did not have astrong a priori positive correlation in terms of delay with otherairports experiencing fluctuations in their delay signals. Inother words, whatever delay signal fluctuations were occurringbetween the affected airport and other airports within the graphwere not significant contributions to the TV, due to the weakcorrelation coefficient edge weights.

VII. SUMMARY AND FUTURE WORK

We analyze the networked behavior of airport delays withinthe US NAS using GSP techniques in order to take intoaccount the underlying airport graph structure and delaycorrelations. In doing so, we compute the TV of our de-lay signals across our graph, and examine their eigenvector(spectral) components. Underpining our spectral analysis isour theoretical work linking together nodal signal correlation,the TV (smoothness) of a graph signal, and unexpected events.

Motivated by the knowledge gap in characterizing nominalor off-nominal disruptions with expected or unexpected airportdelay dynamics at a network-level, our work resulted in twoprimary contributions. The first contribution – a systematicmethod, grounded in theory, to identify days with unexpecteddelay distributions at a network scale – has far-reaching impli-cations for airlines and traffic managers. Through Proposition2, we are able to show that, given historical delay correlations,a new delay signal induced by events such as hurricanes,nor’easters, and airport outages could be classified with ex-plicit probabilistic bounds as expected or unexpected. Enabledby GSP techniques, this classification inherently accounts for

both historical trends and underlying connectivities, allowingfor insightful, network-level interpretations of delay dynamics.Our second contribution stems from our case studies in SectionVI, leading to the observation of fundamental differences inhow delays begin, evolve, and dissipate between differentcategories of disruptive, off-nominal events (e.g. nor’eastersversus Atlantic hurricanes).

In future work, we intend to explore the temporal dynamicsof TV as a disruptive event unfolds; this direction alreadylooks promising from our case study observations, hinting atdeeper connections between TV and airport recovery proce-dures. We also will expand our case study to include all daysclassified as unexpected, enabling us to analyze other typesof disruptions besides nor’easters, hurricanes, airport outages,and NAS-wide high delay days. Finally, we will computeairline-specific graph Laplacians and delay signals, leading tofurther, more detailed operational insights.

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AUTHOR BIOGRAPHIES

Max Z. Li is a PhD Candidate in the Department of Aeronau-tics and Astronautics at the Massachusetts Institute of Technology.His research interests include air traffic flow management, aviationsystems modeling, and applied mathematics, particularly geometricand topological methods. He is a recipient of the 2018 FAA RAISEaward, and a NSF Graduate Research Fellowship.

Karthik Gopalakrishnan is a PhD Candidate in the Departmentof Aeronautics and Astronautics at the Massachusetts Institute ofTechnology. His research interests include network dynamics, controltheory, and modeling of air transportation systems.

Kristyn Pantoja is a PhD Candidate in the Department ofStatistics at the Texas A&M University. Her research interests includemachine learning, statistical education, and design of experiments.

Hamsa Balakrishnan is the Associate Department Head andAssociate Professor of Aeronautics and Astronautics at the Mas-sachusetts Institute of Technology. Her research interests are in thedesign, analysis, and implementation of control and optimizationalgorithms for large-scale cyber-physical infrastructures, with anemphasis on air transportation systems. She is a recipient of theAACC Donald P. Eckman Award (2014), the AIAA Lawrence SperryAward (2012), the CNA Award for Operational Analysis (2012),an NSF CAREER Award (2008), and several best paper awards inICRAT and ATM Seminars.